F05_02SI

Vista previa del material en texto

Related Commercial Resources 
 
Copyright © 2005, ASHRAE
CHAPTER 2
FLUID FLOW
Fluid Properties ............................................................................................................................. 2.1
Basic Relations of Fluid Dynamics ................................................................................................ 2.2
Basic Flow Processes ..................................................................................................................... 2.3
Flow Analysis ................................................................................................................................. 2.6
Noise in Fluid Flow ...................................................................................................................... 2.14
LOWING fluids in HVAC&R systems can transfer heat, mass,Fand momentum. This chapter introduces the basics of fluid
mechanics related to HVAC processes, reviews pertinent flow pro-
cesses, and presents a general discussion of single-phase fluid flow
analysis.
FLUID PROPERTIES
Solids and fluids react differently to shear stress: solids deform
only a finite amount, whereas fluids deform continuously until the
stress is removed. Both liquids and gases are fluids, although the
natures of their molecular interactions differ strongly in both degree
of compressibility and formation of a free surface (interface) in liq-
uid. In general, liquids are considered incompressible fluids; gases
may range from compressible to nearly incompressible. Liquids
have unbalanced molecular cohesive forces at or near the surface
(interface), so the liquid surface tends to contract and has properties
similar to a stretched elastic membrane. A liquid surface, therefore,
is under tension (surface tension).
Fluid motion can be described by several simplified models. The
simplest is the ideal-fluid model, which assumes that the fluid has
no resistance to shearing. Ideal fluid flow analysis is well developed
(e.g., Schlichting 1979), and may be valid for a wide range of appli-
cations.
Viscosity is a measure of a fluid’s resistance to shear. Viscous
effects are taken into account by categorizing a fluid as either New-
tonian or non-Newtonian. In Newtonian fluids, the rate of defor-
mation is directly proportional to the shearing stress; most fluids in
the HVAC industry (e.g., water, air, most refrigerants) can be treated
as Newtonian. In non-Newtonian fluids, the relationship between
the rate of deformation and shear stress is more complicated.
Density
The density ρ of a fluid is its mass per unit volume. The densities
of air and water (Fox et al. 2004) at standard indoor conditions of
20°C and 101.325 kPa (sea level atmospheric pressure) are
Viscosity
Viscosity is the resistance of adjacent fluid layers to shear. A
classic example of shear is shown in Figure 1, where a fluid is
between two parallel plates, each of area A separated by distance Y.
The bottom plate is fixed and the top plate is moving, which induces
a shearing force in the fluid. For a Newtonian fluid, the tangential
The preparation of this chapter is assigned to TC 1.3, Heat Transfer and
Fluid Flow.
ρwater 998 kg m
3
⁄=
ρair 1.21 kg m
3
⁄=
2.
force F per unit area required to slide one plate with velocity V par-
allel to the other is proportional to V/Y:
(1)
where the proportionality factor µ is the absolute or dynamic vis-
cosity of the fluid. The ratio of F to A is the shearing stress τ, and
V/Y is the lateral velocity gradient (Figure 1A). In complex flows,
velocity and shear stress may vary across the flow field; this is
expressed by
(2)
The velocity gradient associated with viscous shear for a simple
case involving flow velocity in the x direction but of varying mag-
nitude in the y direction is illustrated in Figure 1B.
Absolute viscosity µ depends primarily on temperature. For
gases (except near the critical point), viscosity increases with the
square root of the absolute temperature, as predicted by the kinetic
theory of gases. In contrast, a liquid’s viscosity decreases as temper-
ature increases. Absolute viscosities of various fluids are given in
Chapter 39.
Absolute viscosity has dimensions of force × time/length2. At
standard indoor conditions, the absolute viscosities of water and dry
air (Fox et al. 2004) are
Another common unit of viscosity is the centipoise (1 centipoise =
1 g/(s ⋅m) = 1 mPa⋅s). At standard conditions, water has a viscosity
close to 1.0 centipoise.
In fluid dynamics, kinematic viscosity ν is sometimes used in
lieu of the absolute or dynamic viscosity. Kinematic viscosity is the
ratio of absolute viscosity to density:
Fig. 1 Velocity Profiles and Gradients in Shear Flows
Fig. 1 Velocity Profiles and Gradients in Shear Flows
F A⁄ µ V Y⁄( )=
τ µ 
dv
dy
------=
µwater 1.01 mN·s( ) m
2
⁄=
µair 18.1 µN·s( ) m
2
⁄=
1
http://membership.ashrae.org/template/AssetDetail?assetid=42157
2.2 2005 ASHRAE Handbook—Fundamentals (SI)
 
ν = µ /ρ
At standard indoor conditions, the kinematic viscosities of water
and dry air (Fox et al. 2004) are
The stoke (1 cm2/s) and centistoke (1 mm2/s) are common units
for kinematic viscosity.
BASIC RELATIONS OF FLUID DYNAMICS
This section discusses fundamental principles of fluid flow for
constant-property, homogeneous, incompressible fluids and intro-
duces fluid dynamic considerations used in most analyses.
Continuity in a Pipe or Duct
Conservation of mass applied to fluid flow in a conduit requires
that mass not be created or destroyed. Specifically, the mass flow
rate into a section of pipe must equal the mass flow rate out of that
section of pipe if no mass is accumulated or lost (e.g., from leak-
age). This requires that
(3)
where is mass flow rate across the area normal to the flow, v is
fluid velocity normal to the differential area dA, and ρ is fluid den-
sity. Both ρ and v may vary over the cross section A of the conduit.
When flow is effectively incompressible (ρ = constant) in a pipe or
duct flow analysis, the average velocity is then V = (1/A)∫v dA, and
the mass flow rate can be written as
(4)
or
(5)
where Q is the volumetric flow rate.
Bernoulli Equation and Pressure Variation in 
Flow Direction
The Bernoulli equation is a fundamental principle of fluid flow
analysis. It involves the conservation of momentum and energy
along a streamline; it is not generally applicable across streamlines.
Development is fairly straightforward. The first law of thermody-
namics can apply to both mechanical flow energies (kinetic and
potential energy) and thermal energies.
The change in energy content ∆E per unit mass of flowing fluid
is a result of the work per unit mass w done on the system plus the
heat per unit mass q absorbed or rejected:
(6)
Fluid energy is composed of kinetic, potential (because of an eleva-
tion z), and internal (u) energies. Per unit mass of fluid, the energy
change relation between two sections of the system is
(7)
where the work terms are (1) external work EM from a fluid ma-
chine (EM is positive for a pump or blower) and (2) flow work
p/ρ (where p = pressure), and g is the gravitational constant.
νwater 1.01 mm
2
/s=
νair 15.0 mm
2
/s=
m· ρv Ad∫ constant= =
m·
m· ρVA=
Q m· ρ⁄ AV= =
E∆ w q+=
v
2
2
----- gz u+ +⎝ ⎠
⎛ ⎞∆ EM
p
ρ
---
⎝ ⎠
⎛ ⎞∆– q+=
Rearranging, the energy equation can be written as the general-
ized Bernoulli equation:
(8)
The expression in parentheses in Equation (8) is the sum of the
kinetic energy, potential energy, internal energy, and flow work per
unit mass flow rate. In cases with no work interaction, no heat trans-
fer, and no viscous frictional forces that convert mechanical energy
into internal energy, this expression is constant and is known as the
Bernoulli constant B:
(9)
Alternative forms of this relation are obtained through multipli-
cation by ρ or division by g:
(10)
(11)
where γ = ρg is the specific weight or weight density. Note that
Equations (9) to (11) assume no frictional losses.
The units in thefirst form of the Bernoulli equation [Equation
(9)] are energy per unit mass; in Equation (10), energy per unit vol-
ume; in Equation (11), energy per unit weight, usually called head.
In gas flow analysis, Equation (10) is often used, and ρgz is negli-
gible. Equation (10) should be used when density variations occur.
For liquid flows, Equation (11) is commonly used. Identical results
are obtained with the three forms if the units are consistent and flu-
ids are homogeneous.
Many systems of pipes, ducts, pumps, and blowers can be con-
sidered as one-dimensional flow along a streamline (i.e., the varia-
tion in velocity across the pipe or duct is ignored, and local velocity
v = average velocity V ). When v varies significantly across the cross
section, the kinetic energy term in the Bernoulli constant B is
expressed as αV 2/2, where the kinetic energy factor (α > 1)
expresses the ratio of the true kinetic energy of the velocity profile
to that of the average velocity. For laminar flow in a wide rectangu-
lar channel, α = 1.54, and in a pipe, α = 2.0. For turbulent flow in a
duct, α ≈ 1.
Heat transfer q may often be ignored. Conversion of mechanical
energy into internal energy ∆u may be expressed as a loss EL. The
change in the Bernoulli constant (∆B = B2 – B1) between stations 1
and 2 along the conduit can be expressed as
(12)
or, by dividing by g, in the form
(13)
Note that Equation (12) has units of energy per mass while each
term in Equation (13) has units of energy per weight, or head. The
terms EM and EL are defined as positive, where gHM = EM represents
energy added to the conduit flow by pumps or blowers. A turbine or
fluid motor thus has a negative HM or EM. The terms EM and HM (=
EM /g) are defined as positive, and represent energy added to the
v
2
2
----- gz u
P
ρ
----+ + +⎝ ⎠
⎛ ⎞∆ EM= q+
v
2
2
----- gz
P
ρ
----
⎝ ⎠
⎛ ⎞+ + B=
p
ρv
2
2
-------- ρgz+ + ρB=
p
γ
---- ρ
v
2
2g
------ z+ + B
g
---=
p
ρ
---- α
V
2
2
------ gz+ +⎝ ⎠
⎛ ⎞
1
EM EL–+
p
ρ
---- α
V
2
2
------ gz+ +⎝ ⎠
⎛ ⎞
2
=
p
γ
---- α
V
2
2g
------ z+ +⎝ ⎠
⎛ ⎞
1
HM HL–+
p
γ
---- α
V
2
2g
------ z+ +⎝ ⎠
⎛ ⎞
2
=
Fluid Flow 2.3
 
fluid by pumps or blowers. The simplicity of Equation (13) should
be noted; the total head at station 1 (pressure head plus velocity head
plus elevation head) plus the head added by a pump (HM) minus the
head lost due to friction (HL) is the total head at station 2.
Laminar Flow
When real-fluid effects of viscosity or turbulence are included,
the continuity relation in Equation (5) is not changed, but V must be
evaluated from the integral of the velocity profile, using local veloc-
ities. In fluid flow past fixed boundaries, velocity at the boundary is
zero, velocity gradients exist, and shear stresses are produced. The
equations of motion then become complex, and exact solutions are
difficult to find except in simple cases for laminar flow between flat
plates, between rotating cylinders, or within a pipe or tube.
For steady, fully developed laminar flow between two parallel
plates (Figure 2), shear stress τ varies linearly with distance y from
the centerline (transverse to the flow; y = 0 in the center of the chan-
nel). For a wide rectangular channel 2b tall, τ can be written as
(14)
where τw is wall shear stress [b(dp/ds)], and s is the flow direction.
Because velocity is zero at the wall ( y = b), Equation (14) can be
integrated to yield
(15)
The resulting parabolic velocity profile in a wide rectangular
channel is commonly called Poiseuille flow. Maximum velocity
occurs at the centerline ( y = 0), and the average velocity V is 2/3 of
the maximum velocity. From this, the longitudinal pressure drop in
terms of V can be written as
(16)
A parabolic velocity profile can also be derived for a pipe of
radius R. V is 1/2 of the maximum velocity, and the pressure drop
can be written as
(17)
Turbulence
Fluid flows are generally turbulent, involving random perturba-
tions or fluctuations of the flow (velocity and pressure), character-
ized by an extensive hierarchy of scales or frequencies (Robertson
1963). Flow disturbances that are not chaotic but have some degree
of periodicity (e.g., the oscillating vortex trail behind bodies) have
been erroneously identified as turbulence. Only flows involving
random perturbations without any order or periodicity are turbulent;
Fig. 2 Dimension for Steady, Fully Developed Laminar Flow
Equations
Fig. 2 Dimensions for Steady, Fully Developed
Laminar Flow Equations
τ
y
b
----
⎝ ⎠
⎛ ⎞ τw µ 
dv
dy
------= =
v
b
2
y
2
–
2µ
----------------
⎝ ⎠
⎛ ⎞ dp
ds
------=
dp
ds
------
3µV
b
2
------------
⎝ ⎠
⎛ ⎞–=
dp
ds
------
8µV 
R
2
------------
⎝ ⎠
⎛ ⎞–=
the velocity in such a flow varies with time or locale of measurement
(Figure 3).
Turbulence can be quantified statistically. The velocity most
often used is the time-averaged velocity. The strength of turbulence
is characterized by the root mean square (RMS) of the instantaneous
variation in velocity about this mean. Turbulence causes the fluid to
transfer momentum, heat, and mass very rapidly across the flow.
Laminar and turbulent flows can be differentiated using the Rey-
nolds number Re, which is a dimensionless relative ratio of inertial
forces to viscous forces:
(18)
where L is the characteristic length scale and ν is the kinematic vis-
cosity of the fluid. In flow through pipes, tubes, and ducts, the char-
acteristic length scale is the hydraulic diameter Dh, given by
(19)
where A is the cross-sectional area of the pipe, duct, or tube, and Pw
is the wetted perimeter.
For a round pipe, Dh equals the pipe diameter. In general, laminar
flow in pipes or ducts exists when the Reynolds number (based on
Dh) is less than 2300. Fully turbulent flow exists when ReDh >
10 000. For 2300 < ReDh < 10 000, transitional flow exists, and pre-
dictions are unreliable.
BASIC FLOW PROCESSES
Wall Friction
At the boundary of real-fluid flow, the relative tangential velocity
at the fluid surface is zero. Sometimes in turbulent flow studies,
velocity at the wall may appear finite and nonzero, implying a fluid
slip at the wall. However, this is not the case; the conflict results
from difficulty in velocity measurements near the wall (Goldstein
1938). Zero wall velocity leads to high shear stress near the wall
boundary, which slows adjacent fluid layers. Hence, a velocity pro-
file develops near a wall, with velocity increasing from zero at the
wall to an exterior value within a finite lateral distance. 
Laminar and turbulent flow differ significantly in their velocity
profiles. Turbulent flow profiles are flat and laminar profiles are
more pointed (Figure 4). As discussed, fluid velocities of the turbu-
lent profile near the wall must drop to zero more rapidly than those
of the laminar profile, so shear stress and friction are much greater
in turbulent flow. Fully developed conduit flow may be character-
ized by the pipe factor, which is the ratio of average to maximum
(centerline) velocity. Viscous velocity profiles result in pipe factors
of 0.667 and 0.50 for wide rectangular and axisymmetric conduits.
Figure 5 indicates much higher values for rectangular and circular
conduits for turbulent flow. Because of the flat velocity profiles, the
Fig. 3 Velocity Fluctuation at Point in Turbulent Flow
Fig. 3 Velocity Fluctuation at Point in Turbulent Flow
ReL
VL
ν
-------=
Dh
4A
Pw
-------=
2.4 2005 ASHRAE Handbook—Fundamentals (SI)
 
kinetic energy factor α in Equations (12) and (13) ranges from 1.01
to 1.10 for fully developed turbulent pipe flow.
Boundary Layer
The boundary layer is the region close to the wall where wall
friction affects flow. Boundary layer thickness (usually denoted by
δ) is thin compared to downstream flow distance. For external flow
over a body, fluid velocity varies from zero at the wall to a maxi-
mum at distance δ from the wall. Boundary layers are generally
laminar near the start of their formation but may become turbulent
downstream.A significant boundary-layer occurrence exists in a pipeline or
conduit following a well-rounded entrance (Figure 6). Layers grow
from the walls until they meet at the center of the pipe. Near the start
of the straight conduit, the layer is very thin and most likely laminar,
so the uniform velocity core outside has a velocity only slightly
greater than the average velocity. As the layer grows in thickness,
the slower velocity near the wall requires a velocity increase in the
uniform core to satisfy continuity. As flow proceeds, the wall layers
grow (and centerline velocity increases) until they join, after an
entrance length Le. Applying the Bernoulli relation of Equation
(10) to core flow indicates a decrease in pressure along the layer.
Ross (1956) shows that although the entrance length Le is many
diameters, the length in which pressure drop significantly exceeds
that for fully developed flow is on the order of 10 hydraulic diame-
ters for turbulent flow in smooth pipes.
In more general boundary-layer flows, as with wall layer devel-
opment in a diffuser or for the layer developing along the surface of
a strut or turning vane, pressure gradient effects can be severe and
may even lead to boundary layer separation. When the outer flow
velocity (v1 in Figure 7) decreases in the flow direction, an adverse
pressure gradient can cause separation, as shown in the figure.
Downstream from the separation point, fluid backflows near the
wall. Separation is caused by frictional velocity (thus local kinetic
energy) reduction near the wall. Flow near the wall no longer has
energy to move into the higher pressure imposed by the decrease in
Fig. 4 Velocity Profiles of Flow in Pipes
Fig. 4 Velocity Profiles of Flow in Pipes
Fig. 5 Pipe Factor for Flow in Conduits
Fig. 5 Pipe Factor for Flow in Conduits
v1 at the edge of the layer. The locale of this separation is difficult to
predict, especially for the turbulent boundary layer. Analyses verify
the experimental observation that a turbulent boundary layer is less
subject to separation than a laminar one because of its greater
kinetic energy.
Flow Patterns with Separation
In technical applications, flow with separation is common and
often accepted if it is too expensive to avoid. Flow separation may
be geometric or dynamic. Dynamic separation is shown in Figure 7.
Geometric separation (Figures 8 and 9) results when a fluid stream
passes over a very sharp corner, as with an orifice; the fluid gener-
ally leaves the corner irrespective of how much its velocity has been
reduced by friction.
For geometric separation in orifice flow (Figure 8), the outer
streamlines separate from the sharp corners and, because of fluid
inertia, contract to a section smaller than the orifice opening. The
smallest section is known as the vena contracta and generally has
a limiting area of about six-tenths of the orifice opening. After the
vena contracta, the fluid stream expands rather slowly through tur-
bulent or laminar interaction with the fluid along its sides. Outside
the jet, fluid velocity is comparatively small. Turbulence helps
spread out the jet, increases the losses, and brings the velocity dis-
tribution back to a more uniform profile. Finally, downstream, the
velocity profile returns to the fully developed flow of Figure 4. The
entrance and exit profiles can profoundly affect the vena contracta
and pressure drop (Coleman 2004).
Other geometric separations (Figure 9) occur in conduits at sharp
entrances, inclined plates or dampers, or sudden expansions. For
these geometries, a vena contracta can be identified; for sudden
expansion, its area is that of the upstream contraction. Ideal-fluid
theory, using free streamlines, provides insight and predicts con-
traction coefficients for valves, orifices, and vanes (Robertson
1965). These geometric flow separations produce large losses. To
expand a flow efficiently or to have an entrance with minimum
Fig. 6 Flow in Conduit Entrance Region
Fig. 6 Flow in Conduit Entrance Region
Fig. 7 Boundary Layer Flow to Separation
Fig. 7 Boundary Layer Flow to Separation
Fluid Flow 2.5
 
losses, design the device with gradual contours, a diffuser, or a
rounded entrance.
Flow devices with gradual contours are subject to separation that
is more difficult to predict, because it involves the dynamics of
boundary-layer growth under an adverse pressure gradient rather
than flow over a sharp corner. A diffuser is used to reduce the loss
in expansion; it is possible to expand the fluid some distance at a
gentle angle without difficulty, particularly if the boundary layer is
turbulent. Eventually, separation may occur (Figure 10), which is
frequently asymmetrical because of irregularities. Downstream
flow involves flow reversal (backflow) and excess losses. Such sep-
aration is commonly called stall (Kline 1959). Larger expansions
may use splitters that divide the diffuser into smaller sections that
are less likely to have separations (Moore and Kline 1958). Another
technique for controlling separation is to bleed some low-velocity
fluid near the wall (Furuya et al. 1976). Alternatively, Heskested
(1970) shows that suction at the corner of a sudden expansion has a
strong positive effect on geometric separation.
Drag Forces on Bodies or Struts
Bodies in moving fluid streams are subjected to appreciable fluid
forces or drag. Conventionally, the drag force FD on a body can be
expressed in terms of a drag coefficient CD:
(20)
where A is the projected (normal to flow) area of the body. The drag
coefficient CD is a strong function of the body’s shape and angular-
ity, and the Reynolds number of the relative flow in terms of the
body’s characteristic dimension.
Fig. 8 Geometric Separation, Flow Development, and Loss in
Flow Through Orifice
Fig. 8 Geometric Separation, Flow Development, and 
Loss in Flow Through Orifice
Fig. 9 Examples of Geometric Separation Encountered in
Flows in Conduits
Fig. 9 Examples of Geometric Separation 
Encountered in Flows in Conduits
FD CDρA
V 2
2
-------
⎝ ⎠
⎛ ⎞=
For Reynolds numbers of 103 to 105, the CD of most bodies is
constant because of flow separation, but above 105, the CD of
rounded bodies drops suddenly as the surface boundary layer under-
goes transition to turbulence. Typical CD values are given in Table 1;
Hoerner (1965) gives expanded values.
Nonisothermal Effects
When appreciable temperature variations exist, the primary fluid
properties (density and viscosity) may no longer assumed to be con-
stant, but vary across or along the flow. The Bernoulli equation
[Equations (9) to (11)] must be used, because volumetric flow is not
constant. With gas flows, the thermodynamic process involved
must be considered. In general, this is assessed using Equation (9),
written as
(21)
Effects of viscosity variations also appear. In nonisothermal laminar
flow, the parabolic velocity profile (see Figure 4) is no longer valid.
In general, for gases, viscosity increases with the square root of
absolute temperature; for liquids, viscosity decreases with increas-
ing temperature. This results in opposite effects.
For fully developed pipe flow, the linear variation in shear stress
from the wall value τw to zero at the centerline is independent of the
temperature gradient. In the section on Laminar Flow, τ is defined as
τ = ( y/b)τw, where y is the distance from the centerline and 2b is the
wall spacing. For pipe radius R = D/2 and distance from the wall
y = R – r (see Figure 11), then τ = τw(R – y)/R. Then, solving Equa-
tion (2) for the change in velocity yields
(22)
When fluid viscosity is lower near the wall than at the center
(because of external heating of liquid or cooling of gas by heat trans-
fer through the pipe wall), the velocity gradient is steeper near the
wall and flatter near the center, so the profile is generally flattened.
When liquid is cooled or gas is heated, the velocity profile is more
pointed for laminar flow (Figure 11). Calculations for such flows of
Table 1 Drag CoefficientsBody Shape 103 < Re < 2 × 105 Re > 3 × 105
Sphere 0.36 to 0.47 ~0.1
Disk 1.12 1.12
Streamlined strut 0.1 to 0.3 < 0.1
Circular cylinder 1.0 to 1.1 0.35
Elongated rectangular strut 1.0 to 1.2 1.0 to 1.2
Square strut ~2.0 ~2.0
Fig. 10 Separation in Flow in Diffuser
Fig. 10 Separation in Flow in Diffuser
pd
ρ
------
V
2
2
------ gz+ +∫ B=
dv τw R y–( )
Rµ
----------------------- dy 
τw
Rµ
----------
⎝ ⎠
⎛ ⎞– r dr= =
2.6 2005 ASHRAE Handbook—Fundamentals (SI)
 
gases and liquid metals in pipes are in Deissler (1951). Occurrences
in turbulent flow are less apparent than in laminar flow. If enough
heating is applied to gaseous flows, the viscosity increase can cause
reversion to laminar flow.
Buoyancy effects and the gradual approach of the fluid temper-
ature to equilibrium with that outside the pipe can cause consider-
able variation in the velocity profile along the conduit. Colborne and
Drobitch (1966) found the pipe factor for upward vertical flow of
hot air at a Re < 2000 reduced to about 0.6 at 40 diameters from the
entrance, then increased to about 0.8 at 210 diameters, and finally
decreased to the isothermal value of 0.5 at the end of 320 diameters.
FLOW ANALYSIS
Fluid flow analysis is used to correlate pressure changes with
flow rates and the nature of the conduit. For a given pipeline, either
the pressure drop for a certain flow rate, or the flow rate for a certain
pressure difference between the ends of the conduit, is needed. Flow
analysis ultimately involves comparing a pump or blower to a con-
duit piping system for evaluating the expected flow rate.
Generalized Bernoulli Equation
Internal energy differences are generally small, and usually the
only significant effect of heat transfer is to change the density ρ. For
gas or vapor flows, use the generalized Bernoulli equation in the
pressure-over-density form of Equation (12), allowing for the ther-
modynamic process in the pressure-density relation:
(23)
Elevation changes involving z are often negligible and are dropped.
The pressure form of Equation (10) is generally unacceptable when
appreciable density variations occur, because the volumetric flow
rate differs at the two stations. This is particularly serious in friction-
loss evaluations where the density usually varies over considerable
lengths of conduit (Benedict and Carlucci 1966). When the flow is
essentially incompressible, Equation (20) is satisfactory.
Fig. 11 Effect of Viscosity Variation on Velocity
Profile of Laminar Flow in Pipe
Fig. 11 Effect of Viscosity Variation on Velocity
Profile of Laminar Flow in Pipe
dp
ρ
------
1
2
∫– α1
V1
2
2
------ EM+ + α2
V2
2
2
------ EL+=
Example 1. Specify a blower to produce isothermal airflow of 200 L/s
through a ducting system (Figure 12). Accounting for intake and fitting
losses, equivalent conduit lengths are 18 and 50 m and flow is isother-
mal. The pressure at the inlet (station 1) and following the discharge
(station 4), where velocity is zero, is the same. Frictional losses HL are
evaluated as 7.5 m of air between stations 1 and 2, and 72.3 m between
stations 3 and 4.
Solution: The following form of the generalized Bernoulli relation is
used in place of Equation (12), which also could be used:
(24)
The term can be calculated as follows:
(25)
The term can be calculated in a similar manner.
In Equation (24), HM is evaluated by applying the relation between
any two points on opposite sides of the blower. Because conditions at
stations 1 and 4 are known, they are used, and the location-specifying
subscripts on the right side of Equation (24) are changed to 4. Note that
p1 = p4 = p, ρ1 = ρ4 = ρ, and V1 = V4 = 0. Thus,
(26)
so HM = 82.2 m of air. For standard air (ρ = 1.20 kg/m
3), this corre-
sponds to 970 Pa.
The pressure difference measured across the blower (between
stations 2 and 3) is often taken as HM. It can be obtained by calculating
the static pressure at stations 2 and 3. Applying Equation (24) succes-
sively between stations 1 and 2 and between 3 and 4 gives
(27)
where α just ahead of the blower is taken as 1.06, and just after the
blower as 1.03; the latter value is uncertain because of possible uneven
discharge from the blower. Static pressures p1 and p4 may be taken as
zero gage. Thus,
(28)
Fig. 12 Blower and Duct System for Example 1
Fig. 12 Blower and Duct System for Example 1
p1 ρ1g⁄( ) α1 V1
2 2g⁄( ) z1 HM + + +
p2 ρ2g⁄( ) α2 V2
2 2g⁄( ) z2 HL+ + +=
V1
2 2g⁄
A1 π
D
2
-----
⎝ ⎠
⎛ ⎞
2
π
0.250
2
---------------
⎝ ⎠
⎛ ⎞
2
0.0491 m2===
V1 Q A1⁄
0.200
0.0491
---------------- 4.07 m s⁄== =
V1
2 2g⁄ 4.07( )2 2 9.8( )⁄ 0.846 m= =
V2
2 2g⁄
p ρg⁄( ) 0 0.61 HM+ + + p ρg⁄( ) 0 3 7.5 72.3+( )+ + +=
p1 ρg⁄( ) 0 0.61 0+ + + p2 ρg⁄( ) 1.06 0.846×( ) 0 7.5+ + +=
p3 ρg⁄( ) 1.03 2.07×( )+ 0 0+ + p4 ρg⁄( ) 0 3 72.3+ + +=
p2 ρg⁄ 7.8 m of air–=
p3 ρg⁄ 73.2 m of air=
Fluid Flow 2.7
 
The difference between these two numbers is 81 m, which is not the
HM calculated after Equation (24) as 82.2 m. The apparent discrepancy
results from ignoring the velocity at stations 2 and 3. Actually, HM is
(29)
The required blower energy is the same, no matter how it is evalu-
ated. It is the specific energy added to the system by the machine. Only
when the conduit size and velocity profiles on both sides of the
machine are the same is EM or HM simply found from ∆p = p3 – p2.
Conduit Friction
The loss term EL or HL of Equation (12) or (13) accounts for fric-
tion caused by conduit-wall shearing stresses and losses from
conduit-section changes. HL is the loss of energy per unit mass (J/N)
of flowing fluid.
In real-fluid flow, a frictional shear occurs at bounding walls,
gradually influencing flow further away from the boundary. A lat-
eral velocity profile is produced and flow energy is converted into
heat (fluid internal energy), which is generally unrecoverable (a
loss). This loss in fully developed conduit flow is evaluated using
the Darcy-Weisbach equation:
(30)
where L is the length of conduit of diameter D and f is the Darcy-
Weisbach friction factor. Sometimes a numerically different
relation is used with the Fanning friction factor (1/4 of the Darcy
friction factor f ). The value of f is nearly constant for turbulent
flow, varying only from about 0.01 to 0.05.
For fully developed laminar-viscous flow in a pipe, the loss is
evaluated from Equation (17) as follows:
(31)
where Re = VD/v and f = 64/Re. Thus, for laminar flow, the friction
factor varies inversely with the Reynolds number. The value of
64/Re varies with channel shape. A good summary of shape factors
is provided by Incropera and De Witt (2002).
With turbulent flow, friction loss depends not only on flow con-
ditions, as characterized by the Reynolds number, but also on the
roughness height ε of the conduit wall surface. The variation is
complex and is expressed in diagram form (Moody 1944), as shown
in Figure 13. Historically, the Moody diagram has been used to deter-
mine friction factors, but empirical relations suitable for use in mod-
eling programs have been developed. Most are applicable to limited
ranges of Reynolds number and relative roughness. Churchill (1977)
developed a relationship that is valid for all ranges of Reynolds num-
bers, and is more accurate than reading the Moody diagram:
(32a)
(32b)
(32c)
HM p3 ρg⁄( ) α3 V3
2 2g⁄( ) p2 ρg⁄( ) α2 V2
2 2g⁄( )+[ ]–+=
73.2 1.03 2.07×( )+ 7.8 1.06 0.846×( )+–[ ]–=
75.3 6.9–( )– 82.2 m==
HLf f
L
D
------
⎝ ⎠
⎛ ⎞ V
2
2g
--------
⎝ ⎠
⎛ ⎞=
HLf
L
ρg
------
8µV
R
2
-------------
⎝ ⎠
⎛ ⎞ 32LνV
D
2
g
-----------------
64
VD ν⁄
---------------
L
D
------
⎝ ⎠
⎛ ⎞ V
2
2g
--------
⎝ ⎠
⎛ ⎞= = =
f 8
8
ReDh
------------
⎝ ⎠
⎛ ⎞ 12 1
A B+( )
1.5
------------------------+
1 12⁄
=
A 2.457 ln
1
7 ReDh⁄( )
0.9
0.27ε Dh⁄( )+
------------------------------------------------------------------
⎝ ⎠
⎜ ⎟
⎛ ⎞ 16
=
B
37 530
ReDh
-----------------
⎝ ⎠
⎛ ⎞ 16=
Inspection of the Moody diagram indicates that,for high Rey-
nolds numbers and relative roughness, the friction factor becomes
independent of the Reynolds number in a fully rough flow or fully
turbulent regime. A transition region from laminar to turbulent
flow occurs when 2000 < Re < 10 000. Roughness height ε, which
may increase with conduit use, fouling, or aging, is usually tabu-
lated for different types of pipes as shown in Table 2.
Noncircular Conduits. Air ducts are often rectangular in cross
section. The equivalent circular conduit corresponding to the non-
circular conduit must be found before the friction factor can be
determined.
For turbulent flow hydraulic diameter Dh is substituted for D in
Equation (30) and in the Reynolds number. Noncircular duct fric-
tion can be evaluated to within 5% for all except very extreme cross
sections (e.g., tubes with deep grooves or ridges). A more refined
method for finding the equivalent circular duct diameter is given in
Chapter 34. With laminar flow, the loss predictions may be off by a
factor as large as two.
Valve, Fitting, and Transition Losses
Valve and section changes (contractions, expansions and diffus-
ers, elbows, bends, or tees), as well as entrances and exits, distort the
fully developed velocity profiles (see Figure 4) and introduce extra
flow losses that may dissipate as heat into pipelines or duct systems.
Valves, for example, produce such extra losses to control the fluid
flow rate. In contractions and expansions, flow separation as shown
in Figures 9 and 10 causes the extra loss. The loss at rounded
entrances develops as the flow accelerates to higher velocities; this
higher velocity near the wall leads to wall shear stresses greater than
those of fully developed flow (see Figure 6). In flow around bends,
the velocity increases along the inner wall near the start of the bend.
This increased velocity creates a secondary fluid motion in a double
helical vortex pattern downstream from the bend. In all these
devices, the disturbance produced locally is converted into turbu-
lence and appears as a loss in the downstream region. The return of
a disturbed flow pattern into a fully developed velocity profile may
be quite slow. Ito (1962) showed that the secondary motion follow-
ing a bend takes up to 100 diameters of conduit to die out but the
pressure gradient settles out after 50 diameters.
In a laminar fluid flow following a rounded entrance, the
entrance length depends on the Reynolds number:
(33)
At Re = 2000, Equation (33) shows that a length of 120 diameters
is needed to establish the parabolic velocity profile. The pressure
gradient reaches the developed value of Equation (30) in fewer
flow diameters. The additional loss is 1.2V 2/2g; the change in pro-
file from uniform to parabolic results in a loss of 1.0V 2/2g (because
α = 2.0), and the remaining loss is caused by the excess friction. In
turbulent fluid flow, only 80 to 100 diameters following the
rounded entrance are needed for the velocity profile to become
fully developed, but the friction loss per unit length reaches a value
close to that of the fully developed flow value more quickly. After
six diameters, the loss rate at a Reynolds number of 105 is only 14%
above that of fully developed flow in the same length, whereas at
107, it is only 10% higher (Robertson 1963). For a sharp entrance,
the flow separation (see Figure 9) causes a greater disturbance, but
Table 2 Effective Roughness of Conduit Surfaces
Material ε, µm
Commercially smooth brass, lead, copper, or plastic pipe 1.52
Steel and wrought iron 46
Galvanized iron or steel 152
Cast iron 259
Le
D
----- 0.06 Re=
2.8 2005 ASHRAE Handbook—Fundamentals (SI)
fully developed flow is achieved in about half the length required Table 3 lists fitting loss coefficients. These values indicate the
Fig. 13 Relation Between Friction Factor and Reynolds Number
Fig. 13 Relation Between Friction Factor and Reynolds Number
(Moody 1944)
 
for a rounded entrance. In a sudden expansion, the pressure change
settles out in about eight times the diameter change (D2 – D1),
whereas the velocity profile may take at least a 50% greater dis-
tance to return to fully developed pipe flow (Lipstein 1962).
Instead of viewing these losses as occurring over tens or hun-
dreds of pipe diameters, it is possible to treat the entire effect of a
disturbance as if it occurs at a single point in the flow direction. By
treating these losses as a local phenomenon, they can be related to
the velocity by the loss coefficient K:
(34)
Chapter 36 and the Pipe Friction Manual (Hydraulic Institute
1961) have information for pipe applications. Chapter 35 gives
information for airflow. The same type of fitting in pipes and ducts
may yield a different loss, because flow disturbances are controlled
by the detailed geometry of the fitting. The elbow of a small
threaded pipe fitting differs from a bend in a circular duct. For 90o
screw-fitting elbows, K is about 0.8 (Ito 1962), whereas smooth
flanged elbows have a K as low as 0.2 at the optimum curvature.
Loss of section K
V
2
2g
------
⎝ ⎠
⎛ ⎞=
losses, but there is considerable variance. Note that a well-rounded
entrance yields a rather small K of 0.05, whereas a gate valve that is
only 25% open yields a K of 28.8. Expansion flows, such as from
one conduit size to another or at the exit into a room or reservoir, are
not included. For such occurrences, the Borda loss prediction
(from impulse-momentum considerations) is appropriate:
(35)
Expansion losses may be significantly reduced by avoiding or
delaying separation using a gradual diffuser (see Figure 10). For a
diffuser of about 7° total angle, the loss is only about one-sixth of
the loss predicted by Equation (36). The diffuser loss for total angles
above 45 to 60° exceeds that of the sudden expansion, but is mod-
erately influenced by the diameter ratio of the expansion. Optimum
diffuser design involves numerous factors; excellent performance
can be achieved in short diffusers with splitter vanes or suction.
Turning vanes in miter bends produce the least disturbance and loss
for elbows; with careful design, the loss coefficient can be reduced
to as low as 0.1.
Loss at expansion
V1 V2–( )
2
2g
-------------------------
V1
2
2g
------ 1
A1
A2
------–⎝ ⎠
⎛ ⎞
2
= =
Fluid Flow 2.9
 
For losses in smooth elbows, Ito (1962) found a Reynolds num-
ber effect (K slowly decreasing with increasing Re) and a minimum
loss at a bend curvature (bend radius to diameter ratio) of 2.5. At this
optimum curvature, a 45° turn had 63%, and a 180° turn approxi-
mately 120%, of the loss of a 90° bend. The loss does not vary lin-
early with the turning angle because secondary motion occurs.
Note that using K presumes its independence of the Reynolds
number. Some investigators have documented a variation in the loss
coefficient with the Reynolds number. Assuming that K varies with
Re similarly to f, it is convenient to represent fitting losses as adding
to the effective length of uniform conduit. The effective length of a
fitting is then
(36)
where fref is an appropriate reference value of the friction factor.
Deissler (1951) uses 0.028, and the air duct values in Chapter 35 are
based on a fref of about 0.02. For rough conduits, appreciable errors
can occur if the relative roughness does not correspond to that used
when fref was fixed. It is unlikely that the fitting losses involving sep-
aration are affected by pipe roughness. The effective length method
for fitting loss evaluation is still useful.
When a conduit contains a number of section changes or fittings,
the values of K are added to the fL /D friction loss, or the Leff /D of
the fittings are added to the conduit length L /D for evaluating the
total loss HL. This assumes that each fitting loss is fully developed
and its disturbance fully smoothed out before the next section
change. Such an assumption is frequently wrong, and the total loss
can be overestimated. For elbow flows, the total loss of adjacent
bends may be over- or underestimated.The secondary flow pattern
following an elbow is such that when one follows another, perhaps
in a different plane, the secondary flow of the second elbow may
reinforce or partially cancel that of the first. Moving the second
elbow a few diameters can reduce the total loss (from more than
twice the amount) to less than the loss from one elbow. Screens or
perforated plates can be used for smoothing velocity profiles (Wile
1947) and flow spreading. Their effectiveness and loss coefficients
depend on their amount of open area (Baines and Peterson 1951).
Example 2. Water at 20°C flows through the piping system shown in
Figure 14. Each ell has a very long radius and a loss coefficient of
K = 0.31; the entrance at the tank is square-edged with K = 0.5, and
the valve is a fully open globe valve with K = 10. The pipe roughness
is 250 µm. The density ρ = 1000 kg/m3 and kinematic viscosity ν 1.01
mm2/s.
a. If pipe diameter D = 150 mm, what is the elevation H in the tank
required to produce a flow of Q = 60 L/s?
Solution: Apply Equation (13) between stations 1 and 2 in the figure.
Note that p1 = p2, V1 ≈ 0. Assume α ≈ 1. The result is
From Equations (30) and (34), the total head loss is
Leff D⁄ K fref⁄=
Fig. 14 Diagram for Example 2
Fig. 14 Diagram for Example 2
z1 z2– H 12 m– HL
V2
2
2g
------+= =
HL
fL
D
----- ΣK+⎝ ⎠⎛ ⎞
8Q2
π2gD4
----------------=
where L = 102 m, ΣK = 0.5 + (2 × 0.31) + 10 = 11.1, and V 2/2g =
= 8Q2/π2gD4. Then, substituting into Equation (13),
To calculate the friction factor, first calculate Reynolds number and rel-
ative roughness:
Re = = 4Q/(πDv) = 495 150
ε/D = 0.0017
From the Moody diagram or Equation (32), f = 0.023. Then HL =
15.7 m and H = 27.7 m.
b. For H = 22 m and D = 150 mm, what is the flow?
Solution: Applying Equation (13) again and inserting the expression
for head loss gives
Because f depends on Q (unless flow is fully turbulent), iteration is
required. The usual procedure is as follows:
1. Assume a value of f, usually the fully rough value for the given val-
ues of ε and D.
2. Use this value of f in the energy calculation and solve for Q.
3. Use this value of Q to recalculate Re and get a new value of f.
4. Repeat until the new and old values of f agree to two significant
figures.
As shown in the table, the result after two iterations is Q ≈ 0.047 m3/s =
47 L/s.
If the resulting flow is in the fully rough zone and the fully rough
value of f is used as first guess, only one iteration is required.
c. For H = 22 m, what diameter pipe is needed to allow Q = 55 L/s?
Solution: The energy equation in part (b) must now be solved for D
with Q known. This is difficult because the energy equation cannot be
solved for D, even with an assumed value of f. If Churchill’s expression
for f is stored as a function in a calculator, program, or spreadsheet with
an iterative equation solver, a solution can be generated. In this case,
D ≈ 0.166 m = 166 mm. Use the smallest available pipe size greater
than 166 mm and adjust the valve as required to achieve the desired
flow.
Alternatively, (1) guess an available pipe size, and (2) calculate Re,
f, and H for Q = 55 L/s. If the resulting value of H is greater than the
given value of H = 22 m, a larger pipe is required. If the calculated H is
less than 22 m, repeat using a smaller available pipe size.
Control Valve Characterization for Liquids
Control valves are characterized by a discharge coefficient Cd.
As long as the Reynolds number is greater than 250, the orifice
equation holds for liquids:
(37)
where Ao is the area of the orifice opening and ∆p is the pressure
drop across the valve. The discharge coefficient is about 0.63 for
sharp-edged configurations and 0.8 to 0.9 for chamfered or rounded
configurations.
Iteration f Q, m/s Re f
0 0.0223 0.04737 3.98 E + 05 0.0230
1 0.0230 0.04699 3.95 E + 05 0.0230
V2
2 2g⁄
H 12 m 1 fL
D
----- ΣK+ +⎝ ⎠⎛ ⎞
8Q2
π2gD4
----------------+=
VD
v
--------
z1 z2– 10 m
fL
D
----- ΣK 1+ +⎝ ⎠⎛ ⎞
8Q2
π2gD4
----------------+=
Q 
π2gD4 z1 z2–( )
8
fL
D
----- K 1+∑+⎝ ⎠
⎛ ⎞
------------------------------------------=
Q Cd Ao 2 p ρ⁄∆=
2.10 2005 ASHRAE Handbook—Fundamentals (SI)
 
Incompressible Flow in Systems
Flow devices must be evaluated in terms of their interaction with
other elements of the system [e.g., the action of valves in modifying
flow rate and in matching the flow-producing device (pump or
blower) with the system loss]. Analysis is via the general Bernoulli
equation and the loss evaluations noted previously.
A valve regulates or stops the flow of fluid by throttling. The
change in flow is not proportional to the change in area of the valve
opening. Figures 15 and 16 indicate the nonlinear action of valves
in controlling flow. Figure 15 shows flow in a pipe discharging
water from a tank that is controlled by a gate valve. The fitting loss
coefficient K values are from Table 3; the friction factor f is 0.027.
The degree of control also depends on the conduit L /D ratio. For a
relatively long conduit, the valve must be nearly closed before its
high K value becomes a significant portion of the loss. Figure 16
shows a control damper (essentially a butterfly valve) in a duct dis-
charging air from a plenum held at constant pressure. With a long
duct, the damper does not affect the flow rate until it is about one-
quarter closed. Duct length has little effect when the damper is
more than half closed. The damper closes the duct totally at the 90°
position (K = ∞).
Table 3 Fitting Loss Coefficients of Turbulent Flow
Fitting Geometry
Entrance Sharp 0.5
Well-rounded 0.05
Contraction Sharp (D2/D1 = 0.5) 0.38
90° Elbow Miter 1.3
Short radius 0.90
Long radius 0.60
Miter with turning vanes 0.2
Globe valve Open 10
Angle valve Open 5
Gate valve Open 0.19 to 0.22
75% open 1.10
50% open 3.6
25% open 28.8
Any valve Closed ∞
Tee Straight-through flow 0.5
Flow through branch 1.8
K
P∆ ρg⁄
V
2 2g⁄
-------------------
=
Fig. 15 Valve Action in Pipeline
Fig. 15 Valve Action in Pipeline
Flow in a system (pump or blower and conduit with fittings) in-
volves interaction between the characteristics of the flow-producing
device (pump or blower) and the loss characteristics of the pipeline or
duct system. Often the devices are centrifugal, in which case the pres-
sure produced decreases as the flow increases, except for the lowest
flow rates. System pressure required to overcome losses increases
roughly as the square of the flow rate. The flow rate of a given system
is that where the two curves of pressure versus flow rate intersect
(point 1 in Figure 17). When a control valve (or damper) is partially
closed, it increases the losses and reduces the flow (point 2 in Figure
17). For cases of constant pressure, the flow decrease caused by valv-
ing is not as great as that indicated in Figures 15 and 16.
Flow Measurement
The general principles noted (the continuity and Bernoulli equa-
tions) are basic to most fluid-metering devices. Chapter 14 has fur-
ther details.
The pressure difference between the stagnation point (total pres-
sure) and that in the ambient fluid stream (static pressure) is used to
give a point velocity measurement. The flow rate in a conduit is
measured by placing a pitot device at various locations in the cross
section and spatially integrating over the velocity found. A single-
point measurement may be used for approximate flow rate evalua-
tion. When flow is fully developed, the pipe-factor information of
Figure 5 can be used to estimate the flow rate from a centerline
measurement. Measurements can be made in one of two modes.
With the pitot-static tube, the ambient (static) pressure is found from
pressure taps along the side of the forward-facing portion of the
tube. When this portion is not long and slender, static pressure indi-
cation will be low and velocity indication high; as a result, a tube
Fig. 16 Effect of Duct Length on Damper Action
Fig. 16 Effect of Duct Length on Damper Action
Fig. 17 Matching of Pump or Blower to
System CharacteristicsFig. 17 Matching of Pump or Blower to System Characteristics
Fluid Flow 2.11
 
coefficient less than unity must be used. For parallel conduit flow,
wall piezometers (taps) may take the ambient pressure, and the pitot
tube indicates the impact (total pressure).
The venturi meter, flow nozzle, and orifice meter are flow-rate-
metering devices based on the pressure change associated with rel-
atively sudden changes in conduit section area (Figure 18). The
elbow meter (also shown in Figure 18) is another differential pres-
sure flowmeter. The flow nozzle is similar to the venturi in action,
but does not have the downstream diffuser. For all these, the flow
rate is proportional to the square root of the pressure difference
resulting from fluid flow. With area-change devices (venturi, flow
nozzle, and orifice meter), a theoretical flow rate relation is found
by applying the Bernoulli and continuity equations in Equations
(12) and (3) between stations 1 and 2:
(38)
where
∆h = h1 – h2 = p1 – p2/ρg (h = static pressure)
β = d/D = ratio of throat (or orifice) diameter to conduit diameter
The actual flow rate through the device can differ because the
approach flow kinetic energy factor α deviates from unity and
because of small losses. More significantly, the jet contraction of ori-
fice flow is neglected in deriving Equation (38), to the extent that it
can reduce the effective flow area by a factor of 0.6. The effect of all
these factors can be combined into the discharge coefficient Cd:
(39)
Sometimes the following alternative coefficient is used:
(40)
The general mode of variation in Cd for orifices and venturis is
indicated in Figure 19 as a function of Reynolds number and, to a
lesser extent, diameter ratio β. For Reynolds numbers less than 10, the
coefficient varies as .
The elbow meter uses the pressure difference inside and outside
the bend as the metering signal (Murdock et al. 1964). Momentum
analysis gives the flow rate as
(41)
where R is the radius of curvature of the bend. Again, a discharge
coefficient Cd is needed; as in Figure 19, this drops off for lower Rey-
nolds numbers (below 105). These devices are calibrated in pipes
with fully developed velocity profiles, so they must be located far
enough downstream of sections that modify the approach velocity.
Fig. 18 Differential Pressure Flowmeters
Fig. 18 Differential Pressure Flowmeters
Qtheoretical
πd
2
4
--------- 
2 h∆
1 β
4
–
--------------=
Q CdQtheoretical Cd
πd
2
4
--------- 
2 h∆
1 β
4
–
--------------= =
Cd
 1 β
4
–
--------------------
 Re
Qtheoretical
πd
2
4
--------- 
R
2D
------- 2g h∆( )=
Unsteady Flow
Conduit flows are not always steady. In a compressible fluid,
acoustic velocity is usually high and conduit length is rather short,
so the time of signal travel is negligibly small. Even in the incom-
pressible approximation, system response is not instantaneous. If a
pressure difference ∆p is applied between the conduit ends, the fluid
mass must be accelerated and wall friction overcome, so a finite
time passes before the steady flow rate corresponding to the pres-
sure drop is achieved.
The time it takes for an incompressible fluid in a horizontal, con-
stant-area conduit of length L to achieve steady flow may be esti-
mated by using the unsteady flow equation of motion with wall
friction effects included. On the quasi-steady assumption, friction
loss is given by Equation (30); also by continuity, V is constant
along the conduit. The occurrences are characterized by the relation
(42)
where θ is the time and s is the distance in flow direction. Because
a certain ∆p is applied over conduit length L,
(43)
For laminar flow, f is given by Equation (31):
(44)
Equation (44) can be rearranged and integrated to yield the time to
reach a certain velocity:
(45)
and
(46)
Fig. 19 Flowmeter Coefficients
Fig. 19 Flowmeter Coefficients
dV
dθ
-------
1
ρ
-----
⎝ ⎠
⎛ ⎞ dp
 ds
-------
f V 2
2D
---------+ + 0=
dV
dθ
-------
p∆
ρL
------
f V 2
2D
---------–=
dV
dθ
-------
p∆
ρL
------
32µV
ρD
2
-------------– A BV–= =
θ θd∫
Vd
A BV–
-----------------∫ 
1
B
--- A BV–( )ln–= = =
V
p∆
L
-------
D
2
32µ
---------- ⎝ ⎠
⎛ ⎞ 1
ρL
p∆
------
32νθ–
D
2
----------------
⎝ ⎠
⎛ ⎞exp–=
2.12 2005 ASHRAE Handbook—Fundamentals (SI)
 
For long times (θ → ∞), the steady velocity is
(47)
as given by Equation (17). Then, Equation (47) becomes
(48)
where
(49)
The general nature of velocity development for start-up flow is
derived by more complex techniques; however, the temporal varia-
tion is as given here. For shutdown flow (steady flow with ∆p = 0 at
θ > 0), the flow decays exponentially as e–θ.
Turbulent flow analysis of Equation (42) also must be based on
the quasi-steady approximation, with less justification. Daily et al.
(1956) indicate that the frictional resistance is slightly greater than
the steady-state result for accelerating flows, but appreciably less for
decelerating flows. If the friction factor is approximated as constant,
(50)
and for the accelerating flow,
(51)
or
(52)
Because the hyperbolic tangent is zero when the independent
variable is zero and unity when the variable is infinity, the initial
(V = 0 at θ = 0) and final conditions are verified. Thus, for long times
(θ → ∞),
(53)
which is in accord with Equation (30) when f is constant (the flow
regime is the fully rough one of Figure 13). The temporal velocity
variation is then
V∞
p∆
L
-------
D
2
32µ
----------- ⎝ ⎠
⎛ ⎞ p∆
L
-------
R
2
8µ
------- ⎝ ⎠
⎛ ⎞= =
V V∞ 1
ρL
p∆
-------
f∞V∞θ–
2D
--------------------
⎝ ⎠
⎛ ⎞exp–=
f∞
64ν
V∞D
-----------=
Fig. 20 Temporal Increase in Velocity Following
Sudden Application of Pressure
Fig. 20 Temporal Increase in Velocity Following 
Sudden Application of Pressure
dV
dθ
-------
p∆
ρL
------
fV 2
2D
--------– A BV 2–= =
θ
1
 AB
------------- tanh
1–
V 
B
A
-----
⎝ ⎠
⎛ ⎞=
V 
A
B
----- tanh θ AB( )=
V∞ 
A
B
----- 
p∆ ρL⁄
f∞ 2D⁄
------------------ 
p∆
ρL
-------
2D
f∞
-------
⎝ ⎠
⎛ ⎞= = =
(54)
In Figure 20, the turbulent velocity start-up result is compared with
the laminar one, where initially the turbulent is steeper but of the
same general form, increasing rapidly at the start but reaching V∞
asymptotically.
Compressibility
All fluids are compressible to some degree; their density depends
somewhat on the pressure. Steady liquid flow may ordinarily be
treated as incompressible, and incompressible flow analysis is sat-
isfactory for gases and vapors at velocities below about 20 to 40 m/s,
except in long conduits.
For liquids in pipelines, a severe pressure surge or water hammer
may be produced if flow is suddenly stopped. This pressure surge
travels along the pipe at the speed of sound in the liquid, alternately
compressing and decompressing the liquid. For steady gas flows in
long conduits, pressure decrease along the conduit can reduce gas
density significantly enough to increase velocity. If the conduit is
long enough, velocities approaching the speed of sound are possible
at the discharge end, and the Mach number (ratio of flow velocity to
speed of sound) must be considered.
Some compressible flows occur without heat gain or loss (adia-
batically). If there is no friction (conversion of flow mechanical
energy into internal energy), the process is reversible (isentropic), as
well and follows the relationship
where k, the ratio of specific heats at constant pressure and volume,
has a value of 1.4 for air and diatomic gases.
The Bernoulli equation of steady flow, Equation (21), as an inte-
gral of the ideal-fluid equation of motion along a streamline, then
becomes
(55)
where, as in most compressible flow analyses, the elevation terms
involving z are insignificant and are dropped.
For a frictionless adiabatic process, the pressure term has the
form
(56)
Then, between stations 1 and 2 for the isentropic process,
(57)
Equation (57) replaces the Bernoulli equation for compressible
flows and may be appliedto the stagnation point at the front of a
body. With this point as station 2 and the upstream reference flow
ahead of the influence of the body as station 1, V2 = 0. Solving Equa-
tion (57) for p2 gives
(58)
where ps is the stagnation pressure.
V V∞ f∞ V∞θ 2D⁄( )tanh=
p ρ
k
⁄ constant=
k cp cv⁄=
pd
ρ
------
V
2
2
------+∫ constant=
pd
ρ
------
1
2
∫
k
k 1–
-----------
p2
ρ2
-----
p1
ρ1
-----–⎝ ⎠
⎛ ⎞=
p1
ρ1
-----
k
k 1–
-----------
⎝ ⎠
⎛ ⎞ p2
p1
-----
⎝ ⎠
⎛ ⎞
k 1–( ) k⁄
1–
V2
2
V1
2
–
2
------------------+ 0=
ps p2 p1 1
k 1–
2
-----------
⎝ ⎠
⎛ ⎞ ρ1V1
2
kp1
------------+
k k 1–( )⁄
= =
Fluid Flow 2.13
 
Because kp/ρ is the square of acoustic velocity a and Mach num-
ber M = V/a, the stagnation pressure relation becomes
(59)
For Mach numbers less than one,
(60)
When M = 0, Equation (60) reduces to the incompressible flow
result obtained from Equation (9). Appreciable differences appear
when the Mach number of the approaching flow exceeds 0.2. Thus,
a pitot tube in air is influenced by compressibility at velocities over
about 66 m/s.
Flows through a converging conduit, as in a flow nozzle, venturi,
or orifice meter, also may be considered isentropic. Velocity at the
upstream station 1 is negligible. From Equation (57), velocity at the
downstream station is
(61)
The mass flow rate is
(62)
The corresponding incompressible flow relation is
(63)
The compressibility effect is often accounted for in the expan-
sion factor Y:
(64)
Y is 1.00 for the incompressible case. For air (k = 1.4), a Y value
of 0.95 is reached with orifices at p2/p1 = 0.83 and with venturis at
about 0.90, when these devices are of relatively small diameter
(D2/D1 > 0.5).
As p2/p1 decreases, flow rate increases, but more slowly than for
the incompressible case because of the nearly linear decrease in Y.
However, downstream velocity reaches the local acoustic value and
discharge levels off at a value fixed by upstream pressure and den-
sity at the critical ratio:
(65)
At higher pressure ratios than critical, choking (no increase in flow
with decrease in downstream pressure) occurs and is used in some
flow control devices to avoid flow dependence on downstream
conditions.
For compressible fluid metering, the expansion factor Y must be
included, and the mass flow rate is
(66)
ps p1 1
k 1–
2
-----------
⎝ ⎠
⎛ ⎞ M1
2
+
k k 1–( )⁄
=
ps p1
ρ1V1
2
2
------------ 1
M1
4
-------
2 k–
24
-----------
⎝ ⎠
⎛ ⎞ M1
4
…+ + ++=
V2
2k
k 1–
-----------
p1
ρ1
-----
⎝ ⎠
⎛ ⎞ 1
p2
p1
-----
⎝ ⎠
⎛ ⎞
k 1–( ) k⁄
–=
m· V2A2ρ2 =
A2= 
2k
k 1–
----------- p1 ρ1( )
p2
p1
-----
⎝ ⎠
⎛ ⎞
2 k⁄ p2
p1
-----
⎝ ⎠
⎛ ⎞
k 1+( ) k⁄
–
m· in A2ρ 2 p ρ⁄∆ A2 2ρ p1 p2–( )= =
m· Ym· in A2Y 2ρ p1 p2–( )= =
p2
p1
-----
c
2
k 1+
------------
⎝ ⎠
⎛ ⎞ k k 1–( )⁄ 0.53 for air= =
m· CdY
πd
2
4
---------
2ρ∆p
1 β
4
–
--------------=
Compressible Conduit Flow
When friction loss is included, as it must be except for a very
short conduit, incompressible flow analysis applies until pressure
drop exceeds about 10% of the initial pressure. The possibility of
sonic velocities at the end of relatively long conduits limits the
amount of pressure reduction achieved. For an inlet Mach number
of 0.2, discharge pressure can be reduced to about 0.2 of the initial
pressure; for inflow at M = 0.5, discharge pressure cannot be less
than about 0.45p1 (adiabatic) or about 0.6p1 (isothermal).
Analysis must treat density change, as evaluated from the conti-
nuity relation in Equation (3), with the frictional occurrences eval-
uated from wall roughness and Reynolds number correlations of
incompressible flow (Binder 1944). In evaluating valve and fitting
losses, consider the reduction in K caused by compressibility (Bene-
dict and Carlucci 1966). Although the analysis differs significantly,
isothermal and adiabatic flows involve essentially the same pressure
variation along the conduit, up to the limiting conditions.
Cavitation
Liquid flow with gas- or vapor-filled pockets can occur if the
absolute pressure is reduced to vapor pressure or less. In this case,
one or more cavities form, because liquids are rarely pure enough to
withstand any tensile stressing or pressures less than vapor pressure
for any length of time (John and Haberman 1980; Knapp et al. 1970;
Robertson and Wislicenus 1969). Robertson and Wislicenus (1969)
indicate significant occurrences in various technical fields, chiefly
in hydraulic equipment and turbomachines.
Initial evidence of cavitation is the collapse noise of many small
bubbles that appear initially as they are carried by the flow into
higher-pressure regions. The noise is not deleterious and serves as a
warning of the occurrence. As flow velocity further increases or
pressure decreases, the severity of cavitation increases. More bub-
bles appear and may join to form large fixed cavities. The space they
occupy becomes large enough to modify the flow pattern and alter
performance of the flow device. Collapse of cavities on or near solid
boundaries becomes so frequent that, in time, the cumulative impact
causes cavitational erosion of the surface or excessive vibration. As
a result, pumps can lose efficiency or their parts may erode locally.
Control valves may be noisy or seriously damaged by cavitation.
Cavitation in orifice and valve flow is illustrated in Figure 21.
With high upstream pressure and a low flow rate, no cavitation
occurs. As pressure is reduced or flow rate increased, the minimum
pressure in the flow (in the shear layer leaving the edge of the ori-
fice) eventually approaches vapor pressure. Turbulence in this layer
causes fluctuating pressures below the mean (as in vortex cores) and
small bubble-like cavities. These are carried downstream into the
region of pressure regain where they collapse, either in the fluid or
on the wall (Figure 21A). As pressure reduces, more vapor- or gas-
filled bubbles result and coalesce into larger ones. Eventually, a sin-
gle large cavity results that collapses further downstream (Figure
21B). The region of wall damage is then as many as 20 diameters
downstream from the valve or orifice plate.
Fig. 21 Cavitation in Flows in Orifice or Valv
Fig. 21 Cavitation in Flows in Orifice or Valve
2.14 2005 ASHRAE Handbook—Fundamentals (SI)
 
Sensitivity of a device to cavitation is measured by the cavitation
index or cavitation number, which is the ratio of the available pressure
above vapor pressure to the dynamic pressure of the reference flow:
(67)
where pv is vapor pressure, and the subscript o refers to appropriate
reference conditions. Valve analyses use such an index to determine
when cavitation will affect the discharge coefficient (Ball 1957).
With flow-metering devices such as orifices, venturis, and flow noz-
zles, there is little cavitation, because it occurs mostly downstream
of the flow regions involved in establishing the metering action.
The detrimental effects of cavitation can be avoided by operating
the liquid-flow device at high enough pressures. When this is not
possible, the flow must be changed or the device must be built to
withstand cavitation effects. Some materials or surface coatings are
more resistant to cavitation erosion than others, but none is immune.
Surface contours can be designed to delay the onset of cavitation.
NOISE IN FLUID FLOW
Noise in flowing fluids results from unsteady flow fields and can
be at discrete frequencies or broadly distributed over the audible
range. With liquid flow, cavitation results in noise through the col-
lapse of vapor bubbles. Noise in pumps or fittings (e.g., valves) can
be a rattling or sharp hissing sound, which is easily eliminated by
raising the system pressure. With severe cavitation, the resulting
unsteady flow can produce indirect noise from induced vibration of
adjacent parts. See Chapter 47 of the 2003 ASHRAE Handbook—
HVAC Applications for more information on sound control.
The disturbed laminar flow behind cylinders can be an oscillat-
ing motion. Theshedding frequency f of these vortexes is character-
ized by a Strouhal number St = fd/V of about 0.21 for a circular
cylinder of diameter d, over a considerable range of Reynolds num-
bers. This oscillating flow can be a powerful noise source, particu-
larly when f is close to the natural frequency of the cylinder or some
nearby structural member so that resonance occurs. With cylinders
of another shape, such as impeller blades of a pump or blower, the
characterizing Strouhal number involves the trailing-edge thickness
of the member. The strength of the vortex wake, with its resulting
vibrations and noise potential, can be reduced by breaking up flow
with downstream splitter plates or boundary-layer trip devices
(wires) on the cylinder surface.
Noises produced in pipes and ducts, especially from valves and
fittings, are associated with the loss through such elements. The
sound pressure of noise in water pipe flow increases linearly with
pressure loss; broadband noise increases, but only in the lower-
frequency range. Fitting-produced noise levels also increase with
fitting loss (even without cavitation) and significantly exceed noise
levels of the pipe flow. The relation between noise and loss is not
surprising because both involve excessive flow perturbations. A
valve’s pressure-flow characteristics and structural elasticity may
be such that for some operating point it oscillates, perhaps in reso-
nance with part of the piping system, to produce excessive noise. A
change in the operating point conditions or details of the valve
geometry can result in significant noise reduction.
Pumps and blowers are strong potential noise sources. Turbo-
machinery noise is associated with blade-flow occurrences. Broad-
band noise appears from vortex and turbulence interaction with
walls and is primarily a function of the operating point of the
machine. For blowers, it has a minimum at the peak efficiency point
(Groff et al. 1967). Narrow-band noise also appears at the blade-
crossing frequency and its harmonics. Such noise can be very
annoying because it stands out from the background. To reduce this
noise, increase clearances between impeller and housing, and space
impeller blades unevenly around the circumference.
σ
2 po pv–( )
ρVo
2
-------------------------=
Related Comm
REFERENCES
Baines, W.D. and E.G. Peterson. 1951. An investigation of flow through
screens. ASME Transactions 73:467.
Ball, J.W. 1957. Cavitation characteristics of gate valves and globe values
used as flow regulators under heads up to about 125 ft. ASME Trans-
actions 79:1275.
Benedict, R.P. and N.A. Carlucci. 1966. Handbook of specific losses in flow
systems. Plenum Press Data Division, New York.
Binder, R.C. 1944. Limiting isothermal flow in pipes. ASME Transactions
66:221.
Churchill, S.W. 1977. Friction-factor equation spans all fluid flow regimes.
Chemical Engineering 84(24):91-92.
Colborne, W.G. and A.J. Drobitch. 1966. An experimental study of non-
isothermal flow in a vertical circular tube. ASHRAE Transactions
72(4):5.
Coleman, J.W. 2004. An experimentally validated model for two-phase
sudden contraction pressure drop in microchannel tube header. Heat
Transfer Engineering 25(3):69-77.
Daily, J.W., W.L. Hankey, R.W. Olive, and J.M. Jordan. 1956. Resistance
coefficients for accelerated and decelerated flows through smooth tubes
and orifices. ASME Transactions 78:1071-1077.
Deissler, R.G. 1951. Laminar flow in tubes with heat transfer. National Advi-
sory Technical Note 2410, Committee for Aeronautics.
Fox, R.W., A.T. McDonald, and P.J. Pritchard. 2004. Introduction to fluid
mechanics. Wiley, New York.
Furuya, Y., T. Sate, and T. Kushida. 1976. The loss of flow in the conical
with suction at the entrance. Bulletin of the Japan Society of Mechanical
Engineers 19:131.
Goldstein, S., ed. 1938. Modern developments in fluid mechanics. Oxford
University Press, London. Reprinted by Dover Publications, New
York.
Groff, G.C., J.R. Schreiner, and C.E. Bullock. 1967. Centrifugal fan sound
power level prediction. ASHRAE Transactions 73(II):V.4.1.
Heskested, G. 1970. Further experiments with suction at a sudden enlarge-
ment. Journal of Basic Engineering, ASME Transactions 92D:437.
Hoerner, S.F. 1965. Fluid dynamic drag, 3rd ed. Hoerner Fluid Dynamics,
Vancouver, WA.
Hydraulic Institute. 1990. Engineering data book, 2nd ed. Parsippany, NJ.
Incropera, F.P. and D.P. DeWitt. 2002. Fundamentals of heat and mass
transfer, 5th ed. Wiley, New York.
Ito, H. 1962. Pressure losses in smooth pipe bends. Journal of Basic Engi-
neering, ASME Transactions 4(7):43.
John, J.E.A. and W.L. Haberman. 1980. Introduction to fluid mechanics, 2nd
ed. Prentice Hall, Englewood Cliffs, NJ.
Kline, S.J. 1959. On the nature of stall. Journal of Basic Engineering, ASME
Transactions 81D:305.
Knapp, R.T., J.W. Daily, and F.G. Hammitt. 1970. Cavitation. McGraw-Hill,
New York.
Lipstein, N.J. 1962. Low velocity sudden expansion pipe flow. ASHRAE
Journal 4(7):43.
Moody, L.F. 1944. Friction factors for pipe flow. ASME Transactions
66:672.
Moore, C.A. and S.J. Kline. 1958. Some effects of vanes and turbulence in
two-dimensional wide-angle subsonic diffusers. National Advisory
Committee for Aeronautics, Technical Memo 4080.
Murdock, J.W., C.J. Foltz, and C. Gregory. 1964. Performance characteris-
tics of elbow flow meters. Journal of Basic Engineering, ASME Trans-
actions 86D:498.
Olson, R.M. 1980. Essentials of engineering fluid mechanics, 4th ed. Harper
and Row, New York.
Robertson, J.M. 1963. A turbulence primer. University of Illinois (Urbana),
Engineering Experiment Station Circular 79.
Robertson, J.M. 1965. Hydrodynamics in theory and application. Prentice-
Hall, Englewood Cliffs, NJ.
Robertson, J.M. and G.F. Wislicenus, eds. 1969 (discussion 1970). Cavita-
tion state of knowledge. American Society of Mechanical Engineers,
New York.
Ross, D. 1956. Turbulent flow in the entrance region of a pipe. ASME Trans-
actions 78:915.
Schlichting, H. 1979. Boundary layer theory, 7th ed. McGraw-Hill, New
York.
Wile, D.D. 1947. Air flow measurement in the laboratory. Refrigerating
Engineering: 515.
ercial Resources 
http://membership.ashrae.org/template/AssetDetail?assetid=42157
	Main Menu
	SI Table of Contents
	Search
	...entire edition
	...this chapter
	Help
	Fluid Properties 
	Basic Relations of Fluid Dynamics 
	Basic Flow Processes 
	Flow Analysis 
	Noise in Fluid Flow 
	References 
	Tables
	 1 Drag Coefficients 
	 2 Effective Roughness of Conduit Surfaces 
	 3 Fitting Loss Coefficients of Turbulent Flow 
	Figures
	 1 Velocity Profiles and Gradients in Shear Flows 
	 2 Dimension for Steady, Fully Developed Laminar Flow Equations 
	 3 Velocity Fluctuation at Point in Turbulent Flow 
	 4 Velocity Profiles of Flow in Pipes 
	 5 Pipe Factor for Flow in Conduits 
	 6 Flow in Conduit Entrance Region 
	 7 Boundary Layer Flow to Separation 
	 8 Geometric Separation, Flow Development, and Loss in Flow Through Orifice 
	 9 Examples of Geometric Separation Encountered in Flows in Conduits 
	 10 Separation in Flow in Diffuser 
	 11 Effect of Viscosity Variation on Velocity Profile of Laminar Flow in Pipe 
	 12 Blower and Duct System for Example 1 
	 13 Relation Between Friction Factor and Reynolds Number 
	 14 Diagram for Example 2 
	 15 Valve Action in Pipeline 
	 16 Effect of Duct Length on Damper Action 
	 17 Matching of Pump or Blower to System Characteristics 
	 18 Differential Pressure Flowmeters 
	 19 Flowmeter Coefficients 
	 20 Temporal Increase in Velocity Following Sudden Application of Pressure 
	 21 Cavitation in Flows in Orifice or Valve